ESET 111 Week 3: Capacitors and RC Circuits
Chapter 12 Objectives:
Describe characteristics of a capacitor
Analyze series and parallel capacitors
Analyze capacitors in DC circuits
Analyze capacitors in AC circuits
Chapter 15 Objectives:
Determine relationship between current and voltage in an RC circuit
Determine impedance of series, parallel, and series-parallel RC circuits
Analyze series, parallel, and series-parallel RC circuits
Weekly Assignments:
3.1 Discussion: Application of RC Circuits
3.2 Review Assignment: Capacitors and RC Circuits
3.3 Quiz: Capacitors and RC Circuits (Practice)
3.4 Exam: Midterm
3.1 Discussion: Applications of RL Circuits
Capacitive Touch Screens
Run and Start Capacitors
Myth Buster: Capacitors
Capacitor Discharging
Advantages and Disadvantages: Capacitors
Supercapacitors
Troubleshooting Capacitors
Volatile Digital Memory
3.2 Review Assignment: Inductors and RL Circuits
12-1 The Basic Capacitor
12-2 Types of Capacitors
12-3 Series Capacitors
12-4 Parallel Capacitors
12-5 Capacitors in DC Circuits
12-6 Capacitors in AC Circuits
12-7 Capacitor Applications
15-1 The Complex Number System
15-2 Sinusoidal Response of Series RC Circuits
15-3 Impedance of Series RC Circuits
15-4 Analysis of Series RC Circuits
15-5 Impedance and Admittance of Parallel RC Circuits
15-6 Analysis of Parallel RC Circuits
15-7 Analysis of Series-Parallel RC Circuits
15-8 Power in RC Circuits
15-9 Basic Applications
15-10 Troubleshooting
Chapter 15: The Complex Number System
Complex Numbers allow us to do mathematical calculations on phasor quantities in out AC circuits. Numbers are plotted on the complex plane. Numbers one the complex plane can be represented in either polar or rectangular format.
A complex number in rectangular coordinates is written as Re + j Im
A complex number in polar coordinates is written as
Chapter 15: Rectangular to Polar ConversionGeneral
Convert rectangular to coordinates as follows:
The evaluation of the inverse tangent depends upon the quadrant of the angle.
Tan-1 (the principal arctangent) is only defined for -90° to 90°.
If the resultant angle is in the 2nd quadrant, you must add 180° to the result from your calculator.
If the resultant angle is in the 3rd quadrant, you must subtract 180° from the results of your calculator.
We like to express our angles from -180° to 180°
Chapter 15: Rectangular to Polar ConversionFirst Quadrant
Convert the following number to rectangular coordinates:
Given:
10 + j 500
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
510
78.1°
Y-Values010000500Column101000Column201000
Chapter 15: Rectangular to Polar ConversionSecond Quadrant
Convert the following number to rectangular coordinates:
Given:
-122 + j 340
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
361
109.7°
Y-Values0-1220340Column10-122Column20-122
Chapter 15: Rectangular to Polar ConversionThird Quadrant
Convert the following number to rectangular coordinates:
Given:
-222 – j 230
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
320
-134°
Y-Values0-2220-230Column10-222Column20-222
Chapter 15: Rectangular to Polar ConversionFourth Quadrant
Convert the following number to rectangular coordinates:
Given:
416 – j 450
Find:
Polar representation of number
Convert rectangular to coordinates as follows:
613
-47.2°
Y-Values04160-450Column10416Column20416
Chapter 15: Polar to Rectangular ConversionGeneral
Convert rectangular to coordinates as follows:
Drop a perpendicular from the endpoint of the vector to the real axis. This forms a right triangle
Using trigonometry (soh cah toh):
Similarly:
Mag
θ
Y10002200002Y200022000Y300022002
Real Axis
Imaginary Axis
Chapter 15: Convert Radians to Degrees
Convert the following angular value from radians to degrees.
Given:
θ = ½ π radians
Find:
θ (in degrees)
The relationship between degrees and radians can be determined as follows:
Therefore, we can calculate the angle as:
Chapter 15: Convert Degrees to Radians
Convert the following angular value from radians to degrees.
Given:
θ = 260°
Find:
θ (in radians)
The relationship between degrees and radians can be determined as follows:
Therefore, we can calculate the angle as:
Chapter 15: Adding Complex Number
Add the following complex numbers.
Given:
A = 4 + j 3
B = 7 – j 1
Find:
A + B
Complex numbers must be added in rectangular coordinates. To add complex numbers, add the real parts and add the imaginary parts. Note how the negative sign is included with the number.
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 12 + j 15
B = 6 – j 5
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
Polar Multiplication: To multiply numbers in polar coordinates, convert all numbers to polar form, then multiply the magnitudes and add the phase angles.
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 12 + j 15
B = 6 – j 5
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
Rectangular Multiplication: To multiply numbers in rectangular coordinates, use FOIL. Recall that j2 = -1
Chapter 15: Multiplying Complex Numbers
Multiply the following complex numbers.
Given:
A = 9 + j (-4)
B = 27 40°
Find:
A * B
There are two ways to multiply complex numbers: polar multiplication and rectangular multiplication. It is of critical importance that both numbers be expressed in either polar or rectangular coordinates. You may leave the result in either form unless otherwise specified.
For mixed coordinate multiplication, convert both numbers to the same coordinate system and follow the procedures for multiplication, My preference is to multiply in polar coordinates.
Chapter 12: Energy in a Capacitor
What is the energy stored in a 7.2 µF capacitor with a voltage of 8.2 V.
Given:
C = 7.2 µF (capacitance)
V = 8.2 V (voltage)
Find:
W = ½ C V2 (energy)
This formula can be derived by integrating the power over time.
Energy stored in the electric field of a capacitor is found as follows:
Chapter 12: Calculate Series Capacitors
What is the total series capacitance for the following circuit?
Given:
C1 = 5.7 µF
C2 = 11.9 µF
C3 = 17.1 µF
Find:
CT (Total capacitance)
The total capacitance of series capacitors is calculated as follows:
Chapter 12: Voltage Across Series Capacitors Given Charge
Find the Voltage across C2.
Given:
C1 = 7.1 µF
C2 = 10.3 µF
C3 = 9.2 µF
QT = 51 µC
Find:
V2 (voltage across C2)
When a voltage (potential difference) is applied across series capacitors, each capacitor takes on the same charge. In addition, this is the same charge as that across all capacitors:
We also know that for capacitors:
Chapter 15: Voltage Across Series Capacitors Given Vs
Find the Voltage across C2.
Given:
C1 = 17.5 µF
C2 = 8.5 µF
C3 = 10.5 µF
VS = 13.1 V
Find:
VC1 (voltage across C1)
Charge across series capacitors:
We also know that for capacitors:
Therefore:
Chapter 12: Voltage Across Series Capacitors Given Vs
Find the Voltage across C2.
Given:
C1 = 17.5 µF
C2 = 8.5 µF
C3 = 10.5 µF
VS = 13.1 V
Find:
VC1 (voltage across C1)
From the previous slide:
Find CT:
Chapter 12: Calculate Parallel Capacitance
What is the total parallel capacitance for the following circuit?
Given:
C1 = 13.4 µF
C2 = 7.0 µF
C3 = 13.5 µF
Find:
CT (Total capacitance)
The total capacitance of parallel capacitors is calculated as follows:
C=33.9 µF
Chapter 12: Calculate Series-Parallel Capacitor Voltage
What is the voltage between nodes A and B?
Given:
C1 = 147 pF
C2 = 147 pF
C3 = 1470 pF
C4 = 565 pF
C5 = 1470 pF
C6 = 565 pF
VS = 10.3 V
Find:
VAB
First calculate the total capacitance for the voltage divider:
Now perform voltage divider between C5 and C6:
Chapter 12: RC Time Constant
The following circuit shows a capacitor and a resistor in DC circuit. What is the time constant, , for the circuit?
Given:
R = 1.2 kΩ
C = 0.13 µF
Find:
(time constant)
The circuit time constant for and RC circuit determines the rate at which voltage changes in the circuit.
Find the time constant using the following equation:
Chapter 12: Capacitor Charging Value
A charging capacitor will reach what percent of its final value in 0.7 time-constants? Assume the capacitor is initially uncharged.
The capacitor instantaneous voltage is found using:
Where,
VF is the final voltage across the capacitor
vC is the capacitor voltage
is the RC time constant
Start with the capacitor instantaneous voltage:
In our problem
Therefore,
The capacitor reaches 50.3% of its final value in 0.7 time-constants.
Chapter 12: Capacitor Instantaneous Voltage
Find the voltage across the capacitor 22 µs after the switch is closed.
Given:
VS = 39 V
R = 9.5 kΩ
L = 1.8 µF
Find:
VC (22 µs)
First find the time constant using the following equation:
Next determine the final capacitor voltage after the transient response:
Finally, calculate the instantaneous voltage using the following equation:
Chapter 12: Time to Reach Full Charge
How long does it take fort the capacitor to reach full charge once the switch closes? Assume the capacitor is initially uncharged.
Given:
R = 1.4 kΩ
C = 0.19 µF
Find:
Time to full charge
Find the time constant using the following equation:
From the universal exponential curves,
Chapter 12: Calculate Reactance Given Capacitance
What is the value of reactance, XC, for the following circuit given the frequency and capacitance.
Given:
C= 0.039 µH
f = 2 kHz
Find:
XC (reactance)
Capacitive reactance is the opposition to sinusoidal current, expressed in Ohms. The equation for inductive reactance is:
In our problem,
Chapter 12: Series-Parallel Capacitive Voltage
Find the voltage between nodes A and B?
Given:
C1 = 13 µF
C2 = 18 µF
C3 = 9.2 µF
C4 = 9.7 µF
Find:
CAB (Total inductance)
First calculate the series capacitance of C1 and C2:
Next find the parallel combination of C3 and C4:
Chapter 12: Series-Parallel Capacitive Voltage
Find the voltage between nodes A and B?
Given:
C1 = 13 µF
C2 = 18 µF
C3 = 9.2 µF
C4 = 9.7 µF
Find:
CAB (Total inductance)
From the previous slide:
Now find the total capacitance:
Use voltage divider for capacitors,
Chapter 15: Calculate Circuit Impedance
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 130 Ω
XC = 220 Ω
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
In Rectangular Coordinates:
In Polar Coordinates:
Chapter 15: Calculate Circuit Impedance in Rectangular Coordinates
What is the impedance for the following circuit in both rectangular coordinates?
Given:
R = 47 kΩ
C = 2.2 nF
Find:
Z (Impedance)
Impedance for an RL circuit is given by:
Find XC:
f = 100 Hz
f = 500 Hz
f = 2.5 kHz
Impedance for an RL circuit is given by:
Find XC:
f = 100 Hz
f = 500 Hz
f = 2.5 kHz
Chapter 15: Calculate Circuit Impedance – Series Capacitors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 110 kΩ
R2 = 65 kΩ
C1 = 0.047 µF
C2 = 0.047 µF
f = 225 kHz
Find:
Z (Impedance)
Impedance for an RC circuit is given by:
First, find the total series resistance and capacitance:
Chapter 15: Calculate Circuit Impedance – Series Capacitors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R1 = 110 kΩ
R2 = 65 kΩ
C1 = 0.047 µF
C2 = 0.047 µF
f = 225 Hz
Find:
Z (Impedance)
Impedance for an RC circuit is given by:
Next find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
Chapter 15: Calculate Circuit Impedance – Parallel Capacitors
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 40 kΩ
C1 = 100 pF
C2 = 470 pF
f = 20 kHz
Find:
Z (Impedance)
Impedance for an RC circuit is given by:
First, find the total parallel capacitance:
Next find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
Chapter 15: Calculate Impedance of Parallel Circuits
Determine the impedance magnitude and phase angle in polar coordinates for the following circuit.
Given:
R = 2.2 kΩ
XC = 2.0 kΩ
Find:
Z (Impedance)
To find the impedance of a parallel circuit, start by finding the Admittance:
Where:
First, find G and :
Next find Y in rectangular coordinates:
Chapter 15: Calculate Impedance of Parallel Circuits
Determine the impedance magnitude and phase angle in polar coordinates for the following circuit.
Given:
R = 2.2 kΩ
XC = 2.0 kΩ
Find:
Z (Impedance)
From the previous slide:
Convert to polar coordinates:
Find Z in polar coordinates:
You can also convert back to rectangular coordinates:
Chapter 15: Circuit Analysis Series RC Circuit
Determine the total current for the following circuit in polar coordinates?
Given:
R1 = 110 kΩ
R2 = 55 kΩ
C1 = 0.01 µF
C2 = 0.047 µF
f = 200 Hz
V =
Find:
I (Current)
Start by finding the total series resistance and series capacitance: :
Next find the capacitive reactance:
Then find the circuit impedance:
Chapter 15: Circuit Analysis Series RC Circuit
Determine the total current for the following circuit in polar coordinates?
Given:
R1 = 110 kΩ
R2 = 55 kΩ
C1 = 0.01 µF
C2 = 0.047 µF
f = 200 Hz
V =
Find:
I (Current)
From the previous slide:
Use Ohm’s Law to find the current:
Chapter 15: Circuit Parameters and Voltages
What is the impedance for the following circuit in both rectangular and polar coordinates?
Given:
R = 47 Ω
C = 100 µF
f = 20 Hz
V =
Find:
Impedance for an RC circuit is given by:
First, find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
Use Ohm’s Law to find the circuit current:
The voltage across the resistor is given by:
The voltage across the capacitors is given by:
Chapter 15: Circuit Current and Component Voltages
Determine the total current and the voltage across the resistor and capacitors for the following circuit in polar coordinates?
Given:
R = 10 kΩ
C1 = 470 pF
C2 = 220 pF
f = 10 kHz
V =
Find:
Impedance for an RC circuit is given by:
First, find the total parallel capacitance:
Next find the capacitive reactance:
In Rectangular Coordinates:
In Polar Coordinates:
Use Ohm’s Law to find the circuit current:
Chapter 15: Circuit Current and Component Voltages
Determine the total current and the voltage across the resistor and capacitors for the following circuit in polar coordinates?
Given:
R = 10 kΩ
C1 = 470 pF
C2 = 220 pF
f = 10 kHz
V =
Find:
From the previous slide:
The voltage across the resistor is given by:
The voltage across the capacitors is given by:
Notice that :
Chapter 15: RC Lag Circuit Phase Lag
Determine the phase shift between the input and the output voltage for the following RC lag circuit.
Given:
R = 1.5 kΩ
C = 0.27 µF
VS =
f = 250 kHz
Find:
(Phase lag of RC circuit)
The phase lag of an RC circuit is given by the following equation:
Notice the similarity and difference with the phase angle which is given by:
Find capacitive reactance:
Phase Lead:
Phasor diagram showing Vout
Leading Vin (Floyd, 2019)
Chapter 15: Analysis of Parallel RC Circuit
Determine the total circuit current as well as the resistor and capacitor voltages and currents.
Given:
R = 1.5 kΩ
XC = 1.6 kΩ
VS =
Find:
VR, IR, VC, IC, IS
There are at least two approaches to this problem. I will start by finding the branch currents and then adding them:
Find VR and VC:
Use Ohm’s Law to find IR and IL :
Chapter 15: Analysis of Parallel RC Circuit
Determine the total circuit current as well as the resistor and inductor voltages and currents.
Given:
R = 1.5 kΩ
XL = 1.6 kΩ
VS =
Find:
VR, IR, VC, IC, IS
From the last slide:
Use Ohm’s Law to find IL :
Now add the two:
Convert to polar coordinates:
Chapter 16: Power in an RC Circuit
Determine the total, true, and reactive power for the following circuit.
Given:
R = 130Ω
XC = 220 Ω
VS = 10 V
Find:
PTrue, PReactive, PApparent,
Power in an RC circuit is given by:
First find Z:
Next, find I (magnitude only):
Chapter 16: Power in an RC Circuit
Determine the total, true, and reactive power for the following circuit.
Given:
R = 130Ω
XC = 220 Ω
VS = 10 V
Find:
PTrue, PReactive, PApparent,
Find true power:
Find reactive power:
Find apparent power:
From the previous slide:
Power in an RL circuit is given by:
Calculus Based Problem: Derive the Equation for Capacitive Reactance
Derive the expression for capacitive reactance.
Taking derivative:
From this:
Finally,
Start with the equation for inductor voltage:
Voltage is:
Substituting:
Calculus Based Problem: Circuit Analysis in the Time Domain
In the following circuit, determine the current through the resistor (VR) and the current through the capacitor (VC) as well as the total circuit current.
Given:
C = 6.8 µF
R = 220 kΩ
V
Find iR:
Find iS:
3.3 Quiz: Inductors and RC Circuits
Chapter 12 Series Capacitance
Chapter 12 Parallel Capacitance
Chapter 15 Circuit Current and Component Voltages
Chapter 15 Series RC Circuit Analysis
Chapter 15 Parallel RC Circuit Analysis
References
Floyd, Thomas, L. and David M. Buchla. Principles of Electric Circuits. Available from: VitalSource Bookshelf, (10th Edition). Pearson Education (US), 2019.
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